Alternating group coverings of the affine line for characteristic two

A family of étale coverings of the affine line

This note was inspired by a colloquium talk given by S. S. Abhyankar at the Tata Institute, on the work of Abhyankar, Popp and Seiler (see [2]). It was pointed out in this talk that classical modular curves can be used to construct (by specialization) coverings of the affine line in positive characteristic. In this “modular” optic it seemed natural to consider Drinfel’d modular curves for const...

Finite Groups with p-Sylow Coverings

Some combinatorial aspects of finite Hamiltonian groups

In this paper we provide explicit formulas for the number of elements/subgroups/cyclic subgroups of a given order and for the total number of subgroups/cyclic subgroups in a finite Hamiltonian group. The coverings with three proper subgroups and the principal series of such a group are also counted. Finally, we give a complete description of the lattice of characteristic subgroups of a finite H...

Affine Pseudo-planes and Cancellation Problem

We define affine pseudo-planes as one class of Q-homology planes. It is shown that there exists an infinite-dimensional family of non-isomorphic affine pseudo-planes which become isomorphic to each other by taking products with the affine line A1. Moreover, we show that there exists an infinitedimensional family of the universal coverings of affine pseudo-planes with a cyclic group acting as th...

Covers of the affine line in positive characteristic with prescribed ramification

Let k be an algebraically closed field of characteristic p > 0. In this note we consider Galois covers g : Y → P1k which are only branched at t = ∞. We call such covers unramified covers of the affine line. It follows from Abhyankar’s conjecture for the affine line proved by Raynaud ([7]) that such a cover exists for a given group G if and only if G is a quasi-p group, i.e. can be generated by ...